A Benchmarking and Sensitivity Study of the Full Two-Body Gravitational

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A benchmarking and sensitivity study of the full two-body gravitational dynamics of the DART mission target, binary asteroid 65803 Didymos Harrison Agrusa, Derek Richardson, Alex Davis, Eugene Fahnestock, Masatoshi Hirabayashi, Nancy Chabot, Andrew Cheng, Andrew Rivkin, Patrick Michel

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Harrison Agrusa, Derek Richardson, Alex Davis, Eugene Fahnestock, Masatoshi Hirabayashi, et al.. A benchmarking and sensitivity study of the full two-body gravitational dynamics of the DART mission target, binary asteroid 65803 Didymos. Icarus, Elsevier, 2020, 349, pp.113849. 10.1016/j.icarus.2020.113849. hal-02986184

HAL Id: hal-02986184 https://hal.archives-ouvertes.fr/hal-02986184 Submitted on 22 Dec 2020

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A Benchmarking and Sensitivity Study of the Full Two-Body Gravitational Dynamics of the DART Mission Target, Binary Asteroid 65803 Didymos

Harrison F. Agrusaa,∗, Derek C. Richardsona, Alex B. Davisb, Eugene Fahnestockc, Masatoshi Hirabayashid, Nancy L. Chabote, Andrew F. Chenge, Andrew S. Rivkine, Patrick Michelf, and the DART Dynamics Working Group

aDepartment of Astronomy, University of Maryland, College Park, MD 20742, USA bUniversity of Colorado Boulder, 429 UCB, Boulder, CO 80309, USA cJet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA dAuburn University, 39 Davis Hall, Auburn, AL 36849, USA eJohns Hopkins University Applied Physics Laboratory, Laurel MD, 20723, USA fUniversit Cte dAzur, Observatoire de la Cte dAzur, CNRS, Laboratoire Lagrange, 06304 Nice, France

Abstract

NASA’s Double Asteroid Redirection Test (DART) is designed to be the first demonstration of a kinetic impactor for planetary defense against a small-body impact hazard. The target is the smaller component of the binary asteroid 65803 Didymos. We have conducted high-fidelity rigid full two-body simulations of the mutual dynamics of this system in a broad benchmarking exercise to find the best simulation methodologies, and to understand the sensitivity of the system to initial conditions. Due to the non-spherical shapes of the components and their close proximity, the components cannot be treated as point masses and so the dynamics differ significantly from a simple Keplerian orbit, necessitating the use of numerical simulations to fully capture the system’s dynamics. We find that the orbit phase (angular position or true anomaly) of the secondary is highly sensitive to the initial rotation phase of the primary, making prediction of the secondary’s location from numerical simulation challenging. Finally, we show that the DART impact should induce significant free and forced librations on the secondary. If this libration can be measured by ESA’s recently approved follow-up spacecraft, Hera, it may be possible to constrain properties of the secondary’s interior structure. Keywords: Asteroids, dynamics, Rotational dynamics, Celestial mechanics, Near-Earth objects

Preprint submitted to Icarus May 5, 2020 1 1. Introduction

2 The Asteroid Impact & Deﬂection Assessment (AIDA) collaboration is a NASA- and ESA-

3 supported eﬀort to test the capability of a kinetic impactor for hazardous asteroid mitigation.

4 NASA will lead the Double Asteroid Redirection Test (DART) mission, which will achieve a

5 kinetic impact on the secondary (Didymos B) of 65803 Didymos, a near-Earth binary asteroid,

6 in the fall of 2022 (Cheng et al., 2018). ESA will lead Hera, a follow-up mission to rendezvous

7 with Didymos to characterize the system and visible eﬀects of the impact (Michel et al., 2018).

8 The main goal of the DART mission is to demonstrate the kinetic impactor technique by inter-

9 cepting the secondary, causing a change in the binary orbital period that can be measured with

10 ground-based observations.

11 In this work, we present results from a suite of benchmarking simulations conducted by the

12 DART Investigation Team’s Dynamics Working Group to better understand the complex mutual

13 dynamics, to constrain the sensitivity of the simulated Didymos system to initial conditions, and

14 to identify the appropriate numerical methods to fully capture the dynamics. In Section 1.1 we

15 describe the physical and dynamical properties of the Didymos system. Section 2 introduces the

16 four dynamics models used in this study and the initial conditions of the simulation cases. The

17 results are presented in Section 3. Finally, we discuss future work to better understand the Didymos

18 binary in Section 4.

19 1.1. The Didymos System

20 Discovered in 1996, 65803 Didymos is classiﬁed as a near-Earth object and potentially haz-

21 ardous asteroid with a heliocentric semi-major axis of 1.644 au (Alday et al., 1996). In 2003, it

22 was discovered that Didymos is a binary system (Pravec et al., 2003). The binary has a mutual

23 orbit period of Porb ∼ 11.9217 h and a semi-major axis of aorb ∼ 1.19 km (Pravec et al., 2006;

24 Naidu et al., 2020a). According to the binary mean separation and orbital period, Kepler’s third

11 25 law for assumed point masses implies a system mass of Msys ∼ 5.37 × 10 kg.

∗Corresponding author Email address: [email protected] (Harrison F. Agrusa)

2 26 The primary (Didymos A) is ∼780 m across and has an oblate shape and equatorial ridge, and

27 the secondary (Didymos B) is approximately 164 m across and we assume an ellipsoidal shape

28 similar to that of other NEO binary secondaries. Didymos A is a fast rotator, with a spin period

29 of 2.26 h. A polyhedral shape model with 1996 facets was derived by Naidu et al. (2020a) from

30 combined radar and light curve data. In the simulations presented here, Didymos B is assumed to

31 be synchronous (i.e., tidally locked) with its long axis initially aligned with the line of centers. It

32 is also assumed that both bodies are in principal axis rotation and that their spin poles are initially

33 aligned with the binary orbit normal.

34 Table 1 summarizes the relevant physical and dynamical parameters of the Didymos system.

35 These are the nominal system parameters adopted by the DART investigation team at the current

36 time and will be updated throughout the DART mission as new measurements become available.

37 2. Methodology

38 The Didymos binary is an example of the full two-body problem (F2BP), where the rotational

39 and translational dynamics are fully coupled, due to the objects’ irregular shapes and the close

40 proximity of the components. As a result, the system’s dynamical evolution is especially sensitive

41 to the shapes and initial positions and orientations of each component, thus F2BP simulation codes

42 are necessary to fully capture the system’s dynamics.

43 2.1. The Simulation Codes

44 Four diﬀerent codes were tested in this study, each developed by team members at NASA

45 JPL, University of Colorado Boulder (UCB), Auburn University, and the University of Maryland

46 (UMd), respectively. Only some of these codes have oﬃcial names, so we refer to each code by

47 the institution that developed it for simplicity. Brief descriptions of the codes are given below.

48 NASA JPL. The JPL code is based on the formulation of the mutual gravitational potential be-

49 tween two polyhedral bodies developed by Werner and Scheeres (2005). It calculates the mutual

50 gravitational potential and its gradients through a Legendre polynomial series expansion, truncated

51 to a desired order, and integrates the discrete-time Hamiltonian equations of motion using the Lie-

52 Group Variational Integrator (LGVI) developed by Lee et al. (2007). This code was written in C++ 3 Symbol Parameter Value Comments/References

aorb Semi-major Axis 1.19 ± 0.03 km (Naidu et al., 2020a) c : b : a Secondary Axis Ratios 1:1.2:1.56 Assumed, based on other binary systems.

DP Diameter of Primary 780 ± 30 m (Naidu et al., 2020a)

DS Diameter of Secondary 164 ± 18 m Derived from DP and DS/DP

DS/DP Size Ratio 0.21 ± 0.01 (Scheirich and Pravec, 2009)

eorb Binary Orbit eorb < 0.03 Upper limit, assumed zero. (Scheirich and Eccentricity Pravec, 2009)

iorb Binary Orbit 0.0 Assumed. Inclination (λ, β) Mutual Orbit Pole (310◦, −84◦) ± 10◦ Ecliptic coordinates, (Scheirich and Pravec, 2009; Naidu et al., 2020a) 11 Msys Total System Mass (5.37 ± 0.44) × 10 kg Derived via Kepler’s 3rd Law with Porb and aorb.

Porb Binary Orbit Period 11.9217 ± 0.0002 h One possible orbit solution. (Scheirich and Pravec, 2009)

PP Primary Spin Period 2.2600 ± 0.0001 h (Pravec et al., 2006)

PS Secondary Spin Period 11.9217 h Assumed. −3 ρP Primary Bulk Density 2170 ± 350 kg m Derived based on DP and Msys. −3 ρS Secondary Bulk 2170 ± 350 kg m Assumed. Density

Table 1: Physical and dynamical parameters of the Didymos System. These are the current nominal values adopted by the DART investigation team. Because these parameters are constantly being reﬁned by ongoing observations, these are not exactly the same parameters used in this study. The initial conditions of the simulations presented here diﬀer slightly, but remain within the uncertainty bounds given here. (See Table 3 for the simulation initial conditions.)

4 53 and parallelized to run on a cluster computer environment, due to the high computational cost of

54 the potential and gradients evaluation at each timestep.

55 University of Colorado Boulder (UCB). The recently developed UCB code utilizes inertia inte-

56 grals to expand the mutual gravitational potential according to the formalism derived by Hou et al.

57 (2017). This tool, known as the General Use Binary Asteroid Simulator (GUBAS), is now publicly

1 58 available and can easily be run on a single desktop computer. Despite the diﬀerent mathematical

59 formulations for the mutual gravitational potential and its gradients between the JPL and UCB

60 codes, they agree to near-machine precision for the same given expansion order of the mutual

61 potential, since they used the same numerical integrator (LGVI) for the simulations run herein.

62 However, the inertia integral formulation allows for the attitude and mass distribution to be decou-

63 pled and computed separately, which allows for a more computationally eﬃcient implementation

64 and thus faster runtimes. The present study served as a convenient test to conﬁrm that the UCB

65 code does in fact achieve the same result as the JPL code.

66 Auburn University. The Auburn code is a simpliﬁed version of the UCB code. It expands the in-

67 ertia integrals only to second order according to the formulation given by Hirabayashi and Scheeres

68 (2013). The equations of motion are solved with an 8th-order Runge-Kutta scheme. This code can

69 be thought of as evaluating the mutual gravitational potential of the system as if the Didymos A

70 shape model were replaced with a best-ﬁt ellipsoid. Although the Auburn code does not fully cap-

71 ture perturbations due to the asymmetric shape of the primary, it is extremely fast and is a useful

72 reference point to understand the eﬀect of higher-order perturbations due to Didymos A’s shape.

73 University of Maryland (UMd). Unlike the other three codes that represent the primary and sec-

74 ondary as monolithic and homogeneous bodies of some arbitrary polyhedral or ellipsoidal shape,

75 the UMd code treats each body as a rigid aggregate of many spherical particles. The code, called

76 pkdgrav, is a parallel N-body tree code (Richardson et al., 2000; Stadel, 2001). The UMd code

77 uses a primary consisting of ∼3500 particles in order for the average particle diameter (∼42 m) to

1https://github.com/alex-b-davis/gubas

5 Primary Secondary N 3355 3546 −3 ρbulk [g cm ] 2.104 2.104 −3 ρparticle [g cm ] 3.8947 3.7329 ravg [m] 21 4.4

Table 2: Physical parameters for pkdgrav particles. All particles in each respective body are given a uniform particle −3 density in order to achieve the desired bulk density of 2.104 g cm . ravg is the mean particle radius. The particle size distributions for the primary and secondary are sampled from a normal distribution with mean µ = 21 m, standard deviation σ = 4.2 m and µ = 4.4 m, σ = 0.88 m, respectively. The size distributions both have ±1σ cutoﬀs.

78 be within the spatial resolution of the radar shape model (∼50 m). Details of UMd’s representa-

79 tion of each body are shown in Table 2. The translational motion is integrated with a ﬁxed-step

80 second-order leapfrog integrator, while the rotational motion is integrated with a time-adaptive

81 fourth-order Runge-Kutta scheme within each leapfrog step (Richardson et al., 2009). Note that

82 pkdgrav’s k-d tree is not used, so the forces and torques are computed by summing over every

83 particle at every timestep to ensure the highest possible accuracy at the expense of speed.

84 Snapshots of the simulations of Didymos are shown in Fig. 1. The key diﬀerences are that

85 JPL, UCB, and Auburn simulate the full radar-derived Didymos A shape model with an ellipsoidal

86 Didymos B, with the mutual gravitational potential expanded to some desired order of accuracy.

87 The UMd method ﬁlls a volume with randomly packed spherical particles, then carves each body

88 to match the desired shape and computes the mutual gravitational potential of the packed spheres

89 exactly without truncation to any order. Because of the diﬀerent mass representations of each

90 method, there will be inherent variations between these approaches.

91 2.2. Initial Conditions

92 The 11 simulations presented here comprise a small subset of cases that the Dynamics Work-

93 ing Group has studied thus far. These simulations were selected to compare code performance

94 and better understand the system’s sensitivity to uncertainty in its initial state. Using the nom-

95 inal values for the mass of each body and their separation, we computed the initial conditions

96 to approximately put the system on a circular Keplerian orbit. This is considered the “nominal”

97 simulation case. More details on the initial conditions for the nominal case are shown in Table

98 3. We then give these initial conditions slight perturbations to test the system’s sensitivity to the 6 (a) Radar shape model of Didymos A and the assumed ellipsoidal shape of Didymos B. JPL, UCB, and Auburn simulate the full radar-derived primary shape model with the mutual potential expanded to various orders of accuracy.

(b) The UMd representation of the binary, where randomly packed spherical particles ﬁll the shapes of each body and the potential is computed explicitly over every particle.

Figure 1: Representations of the Didymos binary among the diﬀerent codes. Both of these images are a top-down view (i.e. from the mutual orbit north pole) at the start of the simulation.

7 99 initial relative velocity of the secondary and the initial rotation phase of the primary. See Table 4

100 for details of the 11 test cases, along with a schematic in Fig. 2. Each group selected a timestep

101 for their respective code, based on numerical convergence and runtime constraints, with each code

102 conserving energy to one part in a million or better over the entire simulation. Each of the 11

103 test cases was simulated for a total of 150 days of simulation time. All codes modeled the system

104 as two rigid bodies interacting purely through their mutual gravity, with all additional forces or

105 torques such as solar tides, BYORP, or internal dissipation turned oﬀ.

106 Each group output previously agreed-upon state variables at a 1-minute cadence (except for

107 UMd which had hourly outputs due to data storage constraints). With such a high output cadence,

108 we were able to compare both short- and long-term evolution of the binary system with each

109 simulation code. The JPL and UCB codes were run with the mutual gravity expansion set to 4th

110 order; this choice is accurate enough to capture the dynamics with high ﬁdelity while keeping the

111 computation time manageable. UCB also repeated the nominal case with the gravity expansion

112 order set to 8th order to conﬁrm that the choice of 4th order was indeed suﬃcient to accurately

113 model the system. Again, the Auburn code is limited to 2nd order, while the UMd code has no

114 order truncation.

115 3. Results

116 3.1. Code Performance

117 Since each code was run on diﬀerent machines (see acknowledgments) with diﬀerent timesteps

118 and numerical routines, normalized performance comparisons can be troublesome. In Table 5, we

119 simply show the runtimes for the nominal case along with the number of processors and timesteps

120 used by each code. The Auburn code is orders of magnitude faster than the other codes, given

121 its 2nd-order approximation of the mutual potential. This makes it a useful tool for quick tests,

122 however it does not capture higher-order perturbations due to the asymmetric shape of the primary.

123 It should also be noted that the UCB or JPL codes would have similar performance if the mutual

124 potential approximation were set to 2nd order. The UMd code had the longest runtime, due to

125 a combination of its small timestep and requirement to compute the gravitational potential on

8 Parameter Value Notes System Mass 5.276428 × 1011 kg Primary Mass 5.228011 × 1011 kg Secondary Mass 4.841661 × 109 kg Primary Bulk Density 2103.4 kg m−3 Secondary Bulk Density 2103.4 kg m−3 Secondary Axis Lengths a = 103.16 m a is oriented along line of centers at t = 0. b = 79.35 m c = 66.13 m Initial Body Separation 1.18 km Initial Relative Velocity 0.17275 m s−1 Relative velocity of the body centers. Derived to achieve circular Keplerian orbit with period of 11.9216 h. Primary Spin Angular 0.0007723 rad s−1 Equivalent to a 2.26 h spin period. Aligned with mutual Velocity orbit pole. Secondary Spin Angular 0.0001464 rad s−1 Equivalent to a spin period of 11.9216 h, in order to match Velocity the Keplerian orbit period. Aligned with orbit pole.

Table 3: Initial conditions of the nominal simulation.

Name Description nominal The nominal, unperturbed initial state. The initial conditions are calculated using Newton’s version of Kepler’s Third Law to give a circular orbit based on the total system mass and mean separation. posR, negR The initial velocity of the secondary’s barycenter is perturbed by ±0.0005 m/s in the instantaneous orbital radial direction. posT, negT The initial velocity of the secondary’s barycenter is perturbed by ±0.0005 m/s in the instantaneous orbital tangential direction (along-track). posN, negN The initial velocity of the secondary’s barycenter is perturbed by ±0.0005 m/s in the instantaneous orbital normal direction (out-of-plane). ph+1, ph+3, the initial rotation phase of primary shape model is adjusted by rotating ±1 or ±3 degrees ph−1, ph−3 from nominal, around the primary spin pole.

Table 4: Description of the 11 simulation cases.

9 Z axis Y axis

Nominal Trajectory

Side View Top-Down View

posN posT

posR negR X axis X axis Y axis Z axis

negT negN

Figure 2: Schematic of the perturbations to Didymos B. The X, Y, and Z axes are the three principal axes of Didymos A. The nominal case has the long axes of both bodies aligned. The ph±3 and ph±1 cases have Didymos A rotated about its spin axis such that its X axis is pointed ±3 or ±1 degrees away from the direction to Didymos B.

# Processors timestep [s] # timesteps wallclock [h] Auburn 1 60 216,000 ∼several min NASA JPL (4th Order) 512 40 324,000 38.0 UCB (4th Order) 1 40 324,000 5.6 UCB (8th Order) 1 40 324,000 111.55 UMd 4 1.875 6,912,000 702.0

Table 5: Each code’s performance for its nominal run, with a total integration duration of 150 d.

126 a particle-by-particle basis. The 4th-order UCB code oﬀers the best combination of speed and

127 accuracy, as we will see in the following section. Therefore, the Dynamics Working Group has

128 recommended that the UCB code be adopted for future rigid-body dynamics studies related to

129 DART.

130 3.2. The Nominal Case

131 The orbit period, semi-major axis, and eccentricity for each code’s nominal case is shown in

132 Table 6 along with what those values would be if the system were Keplerian. As expected, the

133 NASA JPL and UCB (4th Order) results are nearly identical. They also match closely to the

134 8th-order result, indicating that the 4th-order approximation is capturing the mutual gravitational

10 Orbit Period [h] Semi-Major Axis [km] Eccentricity Auburn 11.8138797 1.1747401 0.0055961390 NASA JPL 11.8062721 1.1743513 0.0059765762 UCB (4th Order) 11.8062824 1.1743520 0.0059765772 UCB (8th Order) 11.8053794 1.1743062 0.0060210983 UMd 11.8211246 1.1750370 0.0051450228 Kepler Orbit 11.9216030 1.1800000 0.0000000000

Table 6: Simulated time-averaged orbital period, semi-major axis, and eccentricity for the nominal case. All codes use exactly the same initial conditions and body masses, so the deviations from a Keplerian orbit and among the codes themselves are due to diﬀerent mass representations of the primary and secondary. Each quantity is rounded to enough decimal points to show deviation between nearly identical numbers.

135 potential with high ﬁdelity. The deviations in the orbit period and semi-major axis are driven

136 by each code’s representation of the mass distribution, and thus the mutual potential, of the two

137 bodies.

138 Figure 3 shows the evolution of the system over 2 d (∼4 orbital periods) as determined by

139 each simulation code for the nominal case. Since the bodies do not follow a precise Keplerian

140 orbit due to their irregular shapes, the orbital eccentricity and inclination are osculating—they

141 are instantaneous values evaluated based on the position and motion of the body centers for each

142 simulation output.

143 Since each group used a numerically converged timestep, the diﬀerences they show in the

144 system’s orbital evolution are attributable to how each code represents the mass distribution of

145 each body. The oscillations in the various orbital elements are small and are driven by the shape

146 perturbations of the primary. The bottom two plots in Fig. 3 show the obliquities of each body,

147 deﬁned as the angle between the body’s spin axis and the mutual orbit pole. Due to its 2nd-order

148 gravity approximation, the Auburn code shows negligible out-of-plane motion, indicating that the

149 small changes in the inclination and obliquities with the other three codes are mainly driven by

150 asymmetries in the primary shape.

151 The NASA JPL and UCB codes at 4th order are indistinguishable in Fig. 3 and the 8th-order

152 version shows almost no appreciable diﬀerence. All codes show qualitative agreement, given their

153 known diﬀerences. Because the UMd code had an output frequency of one hour, the inclination

154 and obliquity plots look artiﬁcially jagged.

11 Auburn (2nd Order) UCB (4th Order) UMd NASA JPL (4th Order) UCB (8th Order) Separation / 1.18 km 1.00 r

0.99 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Eccentricity (osculating) 0.01 e

0.00 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Inclination (osculating), degrees i 0.01 0.00 0.01 − 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Didymos A Obliquity, degrees

εA 0.0000 0.0005 − 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Didymos B Obliquity, degrees 0.025 ε B 0.000 0.025 − 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Time [days]

Figure 3: The nominal case over 2 d (∼4 orbital periods). The NASA JPL and UCB codes show perfect agreement at 4th order, and the UCB 8th-order version matches closely as well. The Auburn code is evaluating the mutual gravitational potential to 2nd order, so it doesn’t capture higher-order eﬀects of the primary’s asymmetric shape on the mutual orbit. The choppy noise in UMd’s plot of Didymos B’s Obliquity is a result of its coarser output cadence. In order to distinguish each curve, this ﬁgure is best viewed in the online color version.

12 Auburn (2nd Order) UCB (4th Order) UMd NASA JPL (4th Order) UCB (8th Order) Separation / 1.18 km 0.996 r

0.995 0 20 40 60 80 100 120 140 Eccentricity (osculating) e 0.006

0.004 0 20 40 60 80 100 120 140 Inclination (osculating), degrees 0.02 i 0.00 0.02 − 0 20 40 60 80 100 120 140 Didymos A Obliquity, degrees 0.0005 εA 0.0000 0.0005 − 0 20 40 60 80 100 120 140 Didymos B Obliquity, degrees 0.025 εB 0.000 0.025 − 0 20 40 60 80 100 120 140 Time [days]

Figure 4: A running average of orbit parameters for the nominal case over 5 months. The first and last 10 d of data are cut off due to the running average using ∼ 10 days worth of data (20 orbital periods). In order to distinguish each curve, this figure is best viewed in the online color version.

155 To study the orbital elements over longer time scales, we take a running average to remove

156 short-term behavior. This is shown for the nominal case over the full 5-month simulation in Fig.

157 4. The long-term evolution is quite stable, with the orbital elements remaining constant or drifting

158 by a very small amount. One apparent eﬀect is that each code has a diﬀerent average semi-

159 major axis and eccentricity. This is again a result of diﬀerent mass representations, which sets

160 the initial mutual potential, and thus the eccentricity and equilibrium separation. This highlights

161 the inﬂuence of the primary’s shape on the orbital properties of the system. In general, the codes

162 agree well, with diﬀerences attributable to their respective representations of each body. Because

163 we expect the Didymos binary to be stable over long periods, our ability to capture long-term

164 stability in its mutual orbit is reassuring.

13 165 3.3. Primary Rotation Phase

166 The only constraints we place on the orientations of the primary and secondary are that their

167 spin axes be initially aligned with the mutual orbit pole. In our nominal case, the primary’s long

168 axis is aligned with the line of centers at t = 0, but this choice is arbitrary. A precise measurement

169 of the primary’s orientation relative to the secondary at a given epoch with ground-based observa-

170 tions prior to the DART impact will be very challenging, so we treat the primary’s initial rotation

171 phase as a free parameter. Therefore, understanding the system’s sensitivity to the initial primary

172 rotation phase is essential to developing methods for predicting the position of the secondary at

173 later times.

174 To test this sensitivity, we varied the initial primary rotation phase with respect to the nominal ◦ ◦ 175 case by ±3 and ±1 . All codes showed a non-negligible dependence on this slight change. Fig. 5

176 shows the orbital phase (angular position of secondary) relative to each code’s respective nominal

177 case. The Auburn code is comparatively insensitive to the initial primary phase since it is only

178 approximating the mass distribution to second order. NASA JPL and UCB have identical results,

179 and UMd is slightly more sensitive to the initial primary phase.

180 If the initial primary rotation phase is altered, the initial mass distribution will be slightly dif-

181 ferent, resulting in a diﬀerent mutual potential and thus a diﬀerent orbital period. Due to the

182 asymmetry of the primary and the binary’s small separation, this is a non-negligible eﬀect, espe-

183 cially if we want to accurately predict the position of the secondary.

184 After determining that the orbit may be sensitive to the initial primary rotation phase, we

185 performed another set of simulations over a wider range of initial rotation phases with the UMd

186 code only. The results in Fig. 6 show that the initial primary rotation phase has a signiﬁcant

187 inﬂuence on the orbital evolution of the secondary. After an integration time of only 30 d, there is ◦ 188 a spread of ∼15 in the relative positions of the secondaries. The DART Investigation Team has a ◦ 189 requirement to predict the orbit phase at the impact epoch to within ±45 60 days prior to launch ◦ 190 and to within ±15 55 days prior to impact (3σ errors). The high sensitivity to Didymos A’s initial

191 rotation phase means that it will likely be impossible to meet this requirement with dynamical

192 simulations, especially considering the uncertainties in the other initial conditions. However, the

193 Observing Working Group should be able to meet this orbital phase prediction requirement through 14 Orbit Phase vs. Initial Primary Rotation Phase All quantitites are relative to their respective nominal run

0.4 Auburn ph+3, ph-3 0.2 ph+1, ph-1 0.0

0.2 − 0.4 − 3 JPL/UCB ph-3 2

1 ph-1

1 ph+1 − 2 − ph+3 3 − 4 UMd ph-3

2 ph-1 Relative Orbit Phase [degrees] 0 ph+1 2 − 4 ph+3 − 0 30 60 90 120 150 Time [days]

Figure 5: Orbit phase relative to each code’s nominal run (the horizontal line). Small differences in the initial primary rotation phase give a slightly different orbital period, making it difficult to predict the position of the secondary at later times. Note the differences in vertical scale between codes.

15 Orbital Phase vs. Initial Primary Rotation Phase Relative to UMd Nominal Case 360

10.0

7.5 270

5.0

Initial Primary 2.5 180 Rotation Phase [degrees]

0.0

2.5 90 −

Relative Orbital Phase5 [degrees] .0 −

0 0 5 10 15 20 25 30 Time [days]

Figure 6: Orbit phase relative to nominal run, for initial primary rotation phases spanning 360◦, using the UMd code. This represents the spread in the possible locations of Didymos B after a ﬁxed interval of time given some random initial primary rotation phase. In order to distinguish each curve, this ﬁgure is best viewed in the online color version.

194 ﬁtting a weighted least-squares model to observed timing of mutual events (Naidu et al., 2020b).

195 3.4. Didymos B Libration

196 Through tidal dissipation, we expect that the mutual orbit has circularized, the secondary is

197 tidally locked, and any libration of the secondary’s spin state has damped to a minimum. So the

198 Didymos system should be in or close to a dynamically relaxed state prior to the DART impact.

199 The impact will nearly instantaneously reduce the instantaneous orbital velocity of the second-ary,

200 decreasing the orbit period and increasing the eccentricity. Therefore, signiﬁcant libration of the

201 secondary should be induced. A libration angle is a measure of the orientation of a satellite’s long

202 axis relative to the line of centers between the two components’ centers of mass. In the following

203 analysis, we show only results from the UCB 4th order code for brevity, although we note that all

204 4 codes show good agreement, given the known diﬀerences among the codes. 16 205 In the coupled spin-orbit problem in which a synchronous, ellipsoidal secondary orbits a spher-

206 ical or point-mass primary with its spin axis aligned with the mutual orbit pole, there are two

207 modes of libration: excited and relaxed. In the decoupled spin-orbit problem, the excited and

208 relaxed modes are analogous to free and forced librations, respectively. See Naidu and Margot

209 (2015) for a detailed discussion on these two libration modes in both the coupled and uncoupled

210 scenarios.

211 The frequency of free libration for a synchronous satellite on a circular orbit is given by,

B − A1/2 ω = n 3 , (1) 0 C

212 where n is the mean motion, and A, B, and C are the three principal moments of inertia of

213 the secondary, where A < B < C (Murray and Dermott, 2000). This mode is analogous to a

214 pendulum’s natural frequency, depending on its length and the gravitational acceleration. This

215 libration mode is thought to be damped away due to tidal friction, especially if the secondary

216 has a rubble-pile structure (Murray and Dermott, 2000; Goldreich and Sari, 2009). However, the

217 forced (relaxed) mode necessarily exists for a synchronous secondary on an eccentric orbit. The

218 secondary will feel a periodic restoring torque, with a frequency equal to the mean motion, due

219 to the misalignment of the long axis with the line of centers, resulting from the orbital angular

220 velocity varying over the course of a single orbit.

221 This picture is complicated when we consider libration in the full two-body problem. The

222 theory on spin-orbit coupling discussed above makes two critical assumptions: 1) that the orbit is

223 ﬁxed (no apsis precession) and 2) that the ellipsoidal secondary is orbiting a spherically symmetric

224 primary (Wisdom, 1987). As a result, we will see some diﬀerences between the classic theory and

225 our simulation results.

226 The nominal DART spacecraft impact is designed to hit the secondary’s center-of-ﬁgure, in

227 a direction nearly opposite its orbital motion at a 15-to-25-degree angle with respect to the or-

228 bital plane (depending on DART’s trajectory), imparting a near-instantaneous change to its orbital

229 velocity without signiﬁcantly altering its spin state(Cheng et al., 2018). This will induce both lon-

230 gitudinal (in-plane) and latitudinal (out-of-plane) librations that will have both relaxed and excited

17 231 components. By changing the mean motion without a matching change in the secondary’s spin

232 rate, we introduce excited (free) libration modes on top of those that exist already (if any). Further,

233 the impact will increase the eccentricity of the system, exciting a stronger relaxed (forced) libration

234 mode. Therefore, studying the resulting libration for the benchmarking cases where we perturb

235 the orbital motion of the secondary reveals the extent to which DART may eﬀect a libration in the

236 secondary. Further, understanding the behavior of induced librations may be an important tool for

237 interpreting the results of the DART impact, if the libration amplitude or frequency is observable.

238 The longitudinal libration for the nominal case is shown in Fig. 7a for the ﬁrst 10 days of the

239 simulation. The distinct beating pattern in the libration is a signature of both excited-mode and

240 relaxed-mode librations (Naidu and Margot, 2015). A Fourier transform of the libration pattern

241 shows the two distinct libration modes (Fig. 7b).

242 The frequency of the relaxed (forced) mode is the frequency at which the orbital angular ve-

243 locity oscillates, which in this case is the epicyclic or radial frequency. A key assumption in the

244 classic spin-orbit problem is that the orbit is ﬁxed (i.e., no apsidal precession), in which case the

245 epicyclic frequency would match the mean motion. However, the oblate shape of the primary and

246 the close orbit of the secondary results in an extremely fast precession of the periapse. In the

247 nominal case, the mean motion diﬀers from the epicyclic frequency by ∼1%, which corresponds ◦ 248 to a precession rate of ∼3.5 per orbit.

249 The excited (free) libration mode has a frequency close to the theoretical prediction given by

250 Eq. 1. These frequencies don’t match perfectly because Eq. 1 assumes a spherically symmetric

251 primary on a ﬁxed orbit. Because this excited libration frequency will depend on the secondary’s

252 moments of inertia in a fashion similar to Eq. 1, it may be possible to infer something about the

253 mass distribution and interior structure from a careful measurement of the libration frequency with

254 Hera.

255 Fig. 8 shows the libration for the nominal case and the 2 cases where the secondary was given

256 an along-track velocity perturbation (posT/negT). The libration amplitude is driven by the initial

257 diﬀerence between the orbital angular velocity and the secondary’s spin rate. The posT case is

258 where the secondary is given a slightly larger initial tangential velocity, so its orbit expands, in-

259 creasing the orbital period to ∼11.911 h, closely matching Didymos B’s initial spin period and 18 4

2 − Libration Angle [degrees]

4 −

0 2 4 6 8 10 Time [days]

(a) Longitudinal libration of UCB nominal case over 10 days. The libration pattern is consistent over the full 150-day simulation.

Epicyclic Frequency Mean Motion Free Libration Frequency

1.0

0.8

0.6

0.4

0.2 Power [arbitrary units]

0.0

0.100 0.095 0.090 0.085 0.080 0.075 0.070 Frequency [1/h]

(b) Fourier transform of longitudinal libration.

Figure 7: The Fourier transform of the longitudinal libration reveals the two libration modes. The relaxed (forced) mode is driven by the epicyclic (radial) frequency, while the excited (free) mode is controlled by the moments of inertia of the secondary. The theoretical free libration frequency doesn’t match the excited mode perfectly because its derivation assumes a spherically symmetric primary.

19 260 thus decreasing the libration amplitude. The negT case is the opposite: a smaller initial tangential

261 velocity shrinks the orbit and shortens the orbital period (∼11.703 h), producing a larger discrep-

262 ancy between the secondary’s initial spin and orbital periods, thus a libration amplitude reaching ◦ 263 ∼8 at its maximum.

264 The perturbation to the secondary’s linear momentum in the posT and negT cases is approx-

265 imately one-half of the momentum carried by the DART spacecraft, so these perturbations are of

266 the same order of magnitude that DART may produce. Since the posT case has a relatively small

267 libration amplitude, we can think of it as being close to the “true” relaxed state of the system (in

268 which the excited libration mode has nearly damped away but relaxed librations persist). Then, the

269 jump from the posT to the negT cases corresponds to a rough conservative estimate of the eﬀect

270 of the DART impact on the libration, when the momentum perturbation to the secondary is ap-

271 proximately equal to the momentum carried by the DART spacecraft. In reality, we would expect

272 the momentum perturbation to the secondary to be considerably larger, due to the contributions of

273 ejecta to the net momentum transfer. Further, if the DART spacecraft impacts several meters oﬀ

274 of center-of-ﬁgure, which terminal guidance simulations at JHU/APL suggest is likely, the torque

275 applied to the secondary will also alter its spin state, nearly instantaneously. The DART terminal

276 guidance system will likely result in an impact location biased toward the illuminated portion of

277 the secondary, which will be the side opposite the primary based on DART’s viewing geometry

278 at the impact epoch. Therefore, such an oﬀ-center impact is likely to reduce the secondary’s an-

279 gular velocity, further increasing the maximum possible libration amplitude. For these reasons,

280 the simulations presented here are a conservative estimate of the possible post-impact libration

281 state of Didymos B, given our current knowledge of the state of the system. It is also important

282 to note, that the libration amplitude and frequency are dependent on Didymos B’s moments of

283 inertia which are computed based on our assumptions of constant density and its ellipsoidal shape.

284 Studying the dependence of Didymos B’s libration on its mass distribution is planned for a future

285 study.

286 Naidu and Margot (2015) show that the libration of a synchronous satellite may be detectable

287 with radar, if the secondary is large enough compared to the primary (DS /DP & 0.2). This thresh-

288 old is barely satisﬁed by the Didymos system and therefore the libration may be measurable with 20 nominal, longitude posT, longitude 7.5 negT, longitude

5.0

2.5

0.0 [degrees] θ 2.5 − 5.0 − 7.5 −

0 1 2 3 4 5 6 7 Time [days]

Figure 8: Longitudinal libration angles a function of time for the nominal run and tangential (along-track) perturbations, using the UCB code. In order to distinguish each curve, this ﬁgure is best viewed in the online color version.

289 radar, given adequate observing conditions, a favorably shaped secondary, and a suﬃciently large

290 momentum transfer.

291 When the secondary is given a normal (out-of-plane) perturbation, we are inducing out-of-

292 plane motion in the secondary and therefore a noticeable, but small, latitudinal libration in the

293 secondary, while the longitudinal libration is eﬀectively unchanged (see Fig. 9). The librations

294 resulting from the benchmarking cases where we apply radial velocity perturbations (posR/negR)

295 and diﬀerent primary rotation phases (ph±3 and ph±1) show almost no sensitivity, so we exclude

296 showing them here.

21 4 nominal, longitude posN, longitude negN, longitude

0 [degrees] θ

2 −

4 − 0 1 2 3 4 5 6 7 Time [days]

(a) Longitudinal libration for out-of-plane perturbations.

nominal, latitude 0.2 posN, latitude negN, latitude

0.1

0.0 [degrees] θ 0.1 −

0.2 − 0 2 4 6 Time [days]

(b) Latitudinal libration for out-of-plane perturbations.

Figure 9: Longitudinal and latitudinal libration vs. time for normal (out-of-plane) perturbations, using the UCB code. The longitudinal libration is insensitive to the normal perturbation as expected, while the small induced latitudinal libration is caused by initial out-of-plane motion of the secondary. In order to distinguish each curve, this ﬁgure is best viewed in the online color version. 22 297 4. Conclusions

298 In this work, we found that the simulation package provided by UC Boulder is well-suited

299 to studying the orbital dynamics of the Didymos system, due to its accuracy and speed. The

300 Dynamics Working Group has recommended the adoption of this code for future dynamics studies

301 in support of DART.

302 The results of this benchmarking study show that: 1) shape perturbations cause a non-negligible

303 deviation from a Keplerian orbit; 2) the orbit phase of the secondary is highly dependent on the

304 initial orientation of the primary; and 3) the system will be highly susceptible to induced librations

305 resulting from the DART impact, which may be measurable from ground-based radar or with Hera.

306 If measurable, Didymos B’s libration may be a useful probe of its internal structure.

307 The ﬁrst two results indicate that predicting the orbital phase of the secondary may not be

308 feasible with numerical simulations, given the uncertainties in the initial conditions and body

309 shapes. However, the Observing Working Group will be able to meet this orbital phase prediction

310 requirement through ﬁtting an analytic model to observed timing of mutual events.

311 We have begun a comprehensive study with the UCB code on the strength and frequency

312 of post-impact librations as a function of the mass distribution of Didymos B and momentum

313 transferred by DART. This study will be used to constrain the range of possible impact outcomes

314 in order to better infer the result of the actual DART experiment.

315 In reality, Didymos A is likely a rubble pile given its shape and fast rotation. A rubble-pile

316 structure may play an important role for the binary dynamics due to processes such as landslides

317 (Hirabayashi et al., 2019) and tidal dissipation. Didymos B may also be a rubble pile in which

318 case its free libration modes will dampen via internal friction. Therefore, we also plan to use pkd-

319 grav with an implementation of a soft-sphere discrete element method to numerically investigate

320 whether this damping may be a noticeable eﬀect over the timescales between the DART and Hera

321 missions.

23 322 Acknowledgments

323 This study was supported in part by the DART mission, NASA Contract # NNN06AA01C

324 to JHU/APL. Some of this research was carried out at the Jet Propulsion Laboratory, California

325 Institute of Technology, under a contract with the National Aeronautics and Space Administration

326 (80NM0018D0004).

327 H.F.A. would like to thank Doug Hamilton for useful discussions. A.B.D. was supported by

328 the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE

329 1650115. Any opinions, ﬁndings, and conclusions or recommendations expressed in this mate-

330 rial are those of the author(s) and do not necessarily reﬂect the views of the National Science

331 Foundation. M.H. acknowledges the Auburn University intramural grant program support. P.M.

332 acknowledges the European Space Agency (ESA) and the French Space Agency (CNES) for sup-

333 port.

334 Some of the simulations herein were carried out on: 1) The University of Maryland Astronomy

335 Department’s YORP cluster, administered by the Center for Theory and Computation; 2) the Stam-

336 pede2 cluster at the Texas Advanced Computing Center (TACC), University of Texas at Austin,

337 funded by NSF award ACI-1134872; and 3) the nexus cluster of the Mission Design and Nav-

338 igation Section at JPL. Raytracing for Fig. 1b was performed using the Persistence of Vision

339 Raytracer (http://povray.org/).

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